The order of the differential equation whose solution is
$y = c_{1} e^{x} + c_{2} sin x + c_{3} cos x$ is

4

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3

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2

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1

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Solution

The correct option is A $3$
$y = c_{1} e^{x} + c_{2} sin x + c_{3} cos x$
$\Rightarrow y_{1} = c_{1} e^{x} + c_{2} cos x - c_{3} sin x$
$\Rightarrow y_{2} = c_{1} e^{x} - c_{2} sin x - c_{3} cos x$
$\Rightarrow y_{2} = c_{1} e^{x} - y + c_{1} e^{x}$
$\Rightarrow y_{2} + y = 2 c_{1} e^{x}$
$\Rightarrow y_{3} + y_{1} = 2 c_{1} e^{x}$
$\Rightarrow y_{3} + y_{1} = y_{2} + y$
$\Rightarrow y_{3} = y_{2} - y_{1} + y$
$\Rightarrow$ order $= 3$ and degree $= 1$ .

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